Lecture on Plotting Parabolas and Understanding Focus, Directrix, and Vertex

Key Concepts

• Plotting Parabolas:
  • Always plot the focus as a point and the directrix as a line.
  • Example given: Directrix is a line at x = 5.

Focus and Directrix

• Focus:
  • A point used to define a parabola.
• Directrix:
  • A line used in conjunction with the focus to define a parabola.

Vertex

• Location:
  • The vertex lies exactly in the middle between the focus and the directrix.
  • In the example, distance between focus and directrix is 3, so vertex is 1.5 from each.
  • Vertex coordinate calculated as (3.5, -3).

Axis of Symmetry

• Properties:
  • Focus and vertex lie on the same axis of symmetry.
  • Axis of symmetry determines direction the parabola opens towards.

Opening Direction of Parabola

• Parabola opens towards the focus.
• Example: Opens to the left, not upwards or to the right.

Formula Derivation

• Standard Form:
  • For horizontal opening, y-coordinate is squared (not x).
  • Formula: ( (y - k)^2 = 4p(x - h) )
• Identifying Parameters:
  • Vertex is at ( (h, k) ).
  • Distance ( p ) from vertex to focus is 1.5 but is negative because it's to the left.

Calculation Example

• Substitute into parabola equation:
  • ( (y + 3)^2 = -6(x - 3.5) )

Important Reminders

• Ensure understanding of parabola shapes and direction.
• Apply formulas correctly by knowing focus, vertex, and direction.
• Plotting initially helps avoid common mistakes.

Questions and Clarity

• Emphasized the importance of understanding graph orientation and formula application.